Is there bus or train service from Nice to St. Paul de Vence? If so, how often does it run?
Yes, there is. Check out Lignes d'Azur. It's time-consuming but can be done. We were there last October, took the train from Nice to Cagnes sur Mer and then found the stop to take the bus (#400, I believe it was) to St. Paul de Vence. However, when the bus went by (!) it said Complet (Full) on the front. We shared a taxi with others who were waiting. Coming home, we took the bus back to Cagnes sur Mer and went to train station only to find that there had been an accident and there wouldn't be any trains in either direction for three hours. So we took the bus back to Nice.
After further reflection, the moral of my story is that you should probably get on the bus in Nice so it won't be full when it gets to another stop. It takes about an hour to make the trip.
Here's the timetable for the 400 bus: https://www.cg06.fr/documents/Import/servir-les-habitants/400.pdf
Thanks for the website for the bus Ligne d'Azur to St. Paul, do you have one for the bus to Eze from Nice?
Here's the main Lignes d'Azur website (click the British flag on the upper right for English): http://www.lignesdazur.com/index.asp
Two lines go from Nice to Eze Village. One, the 112, runs from Nice to Monaco, stopping at Eze in between. It runs only Monday through Saturday: https://www.cg06.fr/documents/Import/servir-les-habitants/112.pdf
The other, the 82, runs every day, but does not go all the way to Monaco: http://www.lignesdazur.com/horaires_ligne/?rub_code=6&keywordsLine=&lign_id=1063